Plato probably wrote his Meno after his first visit with Archytas. In it, Socrates speaks of ‘wise priests and priestesses’ whose authority is reliable, who teach about immortality and reincarnation – a bow to the Pythagoreans. Plato had Socrates attribute to those ‘priests and priestesses’ the idea that because of what we have experienced earlier, much of what we know in this present life is ‘recollection’. This does not seem out of line with what Pythagoras claimed for himself, but Plato had something different in mind that he and his pupils thought was compatible with Pythagorean teaching.
As Plato interpreted the Pythagorean concept of the transmigrating soul, the possibility of escape from the parade of reincarnations lay in ‘becoming just and pious with wisdom’, freeing the soul from fear and the passions and pains of the body. The highest goal was ‘becoming like god’, as Plato phrased it in his Theaetetus. Several centuries later, the pagan neo-Platonist Porphyry (Pythagoras’ biographer) listed Hercules, Pythagoras, and Jesus among those who had succeeded in this ultimate achievement of ‘becoming like god’.
‘Recollection’, for Plato, did not however mean memories of past lives. Instead, it was the mysterious, innate, a priori knowledge that humans seem to possess, that cannot be explained by what one has learned in one’s present life. Plato did not imply that anyone could recall acquiring this knowledge. As a demonstration, he used the geometric exercise that some believe reflected Pythagoras’ proof of the Pythagorean theorem.
In this scene from Plato’s Meno, Socrates and Meno are discussing a figure Socrates has drawn in the sand, a four-foot square. The task is to double the size of the square. Socrates intends to demonstrate that innate knowledge – not of the correct answer but of the underlying geometry that will lead to the correct answer – lies hidden in Meno’s slave boy, waiting to be reawakened. Socrates is acting as a sort of midwife.
SOCRATES (to Meno): Pay attention and I’ll show you what he can discover by seeking the truth in my company, though I will only ask him questions, not teach him. Catch me if I seem to instruct him or explain instead of only asking him questions. (Socrates erases previous figures in the sand and draws a four-foot square.)
Tell me, young man. Is this our square of four feet? Do you understand?
BOY: It is.
SOCRATES: Can we add another square, equal to it, like this? (Draws.)
BOY: Very well.
SOCRATES: And here a third square, equal to each of the others? (Draws.)
BOY: Yes.
SOCRATES: And now we can fill in this corner with another square? (Draws.)
BOY: Yes.
SOCRATES: So now we have four equal squares?
BOY: We do.
SOCRATES: And how many times the size of our first square is this whole figure?
BOY: Four times.
SOCRATES: But we wanted one two times as large, remember?
BOY: Yes.
SOCRATES: Now, if we draw lines from corner to corner, like this, do they cut each of those squares in half? (Draws.)
BOY: They do.
SOCRATES: And those four new lines create a new, central square?
BOY: Yes.
SOCRATES: Now think, how big is this new central square?
BOY: I do not understand.
SOCRATES: Again, here are our four squares. Does not each new line cut off half of one of them?
BOY: Yes.
SOCRATES: And how many of those halves are now included in this central square?
BOY: Four of them.
SOCRATES: How many are in each of the original squares?
BOY: Two.
SOCRATES: And the relationship of four to two is?
BOY: Double.
SOCRATES: So, how big is this figure?
BOY: Eight feet.
SOCRATES: On what base?
BOY: This. (He points to one of the diagonal lines.)
SOCRATES: You mean the line going from corner to corner of the square of four feet?
BOY: Yes.
SOCRATES: Clever men call such a line a ‘diagonal’; so, using that name, is it your conclusion that the square on the diagonal of the original square is double its area?
BOY: Most certainly, Socrates.
SOCRATES: Well, Meno, do you think he has expressed any opinion that was not his own?
MENO: No, they were all his own.
SOCRATES: Yet, as we agreed a few minutes ago, he did not know?
MENO: That is true.
SOCRATES: So these opinions were somewhere in him, were they not?
MENO: They were.
SOCRATES: So a man who does not know nevertheless has within himself true opinions about things that he does not know?
MENO: It appears so.
SOCRATES: These opinions, at present, have been newly stirred up and have a dream-like quality. But put the same questions to him repeatedly in various ways and you can see that in the end his knowledge about these things will be as good as anyone’s.
MENO: Very likely.
SOCRATES: He will not have been taught. Only questioned. He will find the knowledge in himself.
MENO: Yes.
SOCRATES: And is such finding of knowledge within oneself not recollection?
MENO: Certainly.
SOCRATES: Then he has either at some time acquired the knowledge he now has, or he has always had it. If he always had it, he must always have known; but if he acquired it at some previous time, it cannot have been in this present lifetime, unless somebody has taught him geometry. It will be the same for him with all geometry, and every other subject. Has anyone taught him all these? You would know. He has been raised in your household.
MENO: I know that nobody has taught him.
SOCRATES: Yet he does have these opinions, doesn’t he?
MENO: That is indisputable, Socrates.
SOCRATES: If he did not acquire them in this life, is it not clear that he possessed them and had learned them during some other time?
MENO: It seems so.
SOCRATES: A time when he was not in human shape.
MENO: Yes.
A modern attorney would probably object that Socrates was ‘leading the witness’. But Plato was not talking about knowledge the boy had hidden somewhere in his mind because he had witnessed it or been taught it in a previous life: the date of an event or the length of a road – knowledge of the changeable world. Plato meant inborn knowledge of truths that do not change – universal and immutable truths of the Forms, in this case truths of geometry. The point of Plato’s lesson scene was that at each stage of questioning, the boy knew whether what Socrates was suggesting was correct. Such recollection of the ‘eternal Forms’ came not from past lives at all but from experiences of the disembodied soul.
Many who first encounter proofs in a setting other than a smotheringly dry presentation are struck by this deep, mysterious sense of recognition of something they already knew. Indeed there are truths that have been ‘rediscovered’ time and time again (the Pythagorean theorem may be one of them) by ancient people and by more recent individuals who were unaware they were repeating a former discovery. Socrates’ demonstration was an extremely Pythagorean lesson, for it united the two Pythagorean themes: the immortality of the soul and the mathematical structure of the world.
Other dialogues and his Republic show that Plato’s mind was much taken up with the doctrines of recollection, reincarnation, and immortality. His Phaedo ends shortly after Socrates’ death, with Phaedo pausing on his journey home from Athens in a Pythagorean community in Phlius to tell Echecrates and other Pythagoreans about the philosopher’s last words. In a discussion centring on immortality and reincarnation, Phaedo repeats Socrates’ quote from an Orphic poem that Socrates had thought spoke of philosophy�
��s power to raise one to the level of the gods. In his Phaedrus, Plato wrote that human ‘love’ was recollection of the experience of Beauty as an eternal Form.
In his ‘Myth of Er’, at the end of The Republic, Plato most clearly revealed his belief in reincarnation, although, true to his doctrine that knowledge of ultimate truth is unattainable, he used the term ‘myth’ to indicate that he could not vouch for the absolute truth of the lessons it taught. In the ‘myth’ he imagined what happens when one life has ended and the next has not yet begun: Each soul chooses what it will be in the next life. Choices include ‘lives of all living creatures, as well as of all conditions of men’. Orpheus chooses to be a swan so as not to be born of a woman – for frenzied Bacchic women had torn him apart in a former life – while a soul who has lived previously as a swan chooses to be a man. The harmony of the spheres was also on Plato’s mind. The souls see a vision, a magnificent model of the cosmos. On each of the circles in which the planets and other bodies orbit stands ‘a Siren, who was carried round with its movement, uttering a single sound on one note, so that all the eight made up the concords of a single scale’. Though Earth, in Plato’s cosmos, sat dead centre, and there was no central fire or counter-earth, the ‘Myth of Er’ was suffused with Pythagorean ideas.
When the members of Plato’s Academy before and after his death in 348/347 B.C. thought about Pythagoras and called themselves Pythagorean, they had in mind mainly Pythagoras as seen through Plato’s eyes. However, to say that Pythagoras was reinvented as a ‘late Platonist’, as some scholars insist, is to be too glib and overconfident about where to draw the lines between original Pythagorean thought, Pythagorean thought shortly after Pythagoras’ death, Archytas, Plato, and Plato’s pupils, some of whom attributed their own ideas to more ancient Pythagoreans and even to Pythagoras. As time passed, the line between Platonism and what called itself Pythagorean became increasingly difficult to discern. Eventually the two were indistinguishable.
[1]‘Diatonic’ refers to the scales now known as major and minor scales.
[2]Plato was not the first to think of the planets moving on rings. Anaximander’s cosmos involved huge wheels, whose hollow rims were filled with fire. The Sun, Moon, stars and planets were glimpses of this fire, showing through at openings in the wheel rims. Similar ideas had surfaced elsewhere as well. After Plato, the idea was taken up by his pupil Eudoxus, who responded to Plato’s challenge to produce an analysis that would account for the appearances in the heavens with an explanation along the lines introduced by the Pythagoreans, involving a combination of movements of the sphere of stars and the planets. Eudoxus did this with a system not of concentric rings but of concentric spheres, and that was adopted by Aristotle and would dominate astronomy until the time of Tycho Brahe and Johannes Kepler.
[3]Kepler discovered other regular solids, the ‘hedgehog’, for example, but they did not have all the characteristics of the original five.
[4]‘The Academy’ also refers to the men associated with this school after Plato’s lifetime, including his successors as scholarch elected for life by a majority vote of the members. Aristotle was also associated with the Academy, first as a pupil and later as a teacher. In several transformations, still claiming descent from the original, the Academy lasted until the sixth century A.D. as a centre of Platonism and neo-Platonism.
[5]The writer Richard E. Rubenstein put it succinctly: ‘Plato did not hate the world, it simply reminded him of a better place’ (Richard E. Rubenstein, Aristotle’s Children: How Christians, Muslims, and Jews Rediscovered Ancient Wisdom and Illuminated the Dark Ages [New York: Harcourt, 2003]).
CHAPTER 10
From Aristotle to Euclid
Fourth Century B.C.
While most scholars were content to view Pythagorean teachings through Plato’s eyes and not eager to differentiate between Plato’s philosophy and the thinking of pre-Platonic Pythagoreans, one person was still curious. That was Aristotle. Born in 384, he was two generations younger than Plato and at age seventeen had come to Athens to study at Plato’s Academy. Plato was away at the time, on one of his jaunts to Sicily. Twenty years later, when Plato died at age eighty in 348, Aristotle was only thirty-seven and, perhaps because of his youth, was not chosen to succeed Plato as scholarch of the Academy. Instead, though by then hardly anyone failed to recognise that Aristotle was one of the most gifted men around, Plato’s nephew Speusippus got the job. Aristotle left Athens and eventually returned to found his own school, the Lyceum. His debt to Plato was clear throughout his work, but so was the fact that the two disagreed in significant ways. Aristotle was not happy with Plato’s concept of Forms. Plato thought the world as humans knew it was only an undependable reflection of a real world that humans could never know. Aristotle, by contrast, believed that the world humans perceive is the real world. He highly valued what could be learned about nature through use of the human senses, and what could be extrapolated from those perceptions. It would not have displeased Aristotle to find that Plato’s teachings were at least in part derivative of the Pythagoreans. In his Metaphysics, in a passage following his description of Pythagorean philosophies, Aristotle looked down his nose at Plato and invited his readers to do the same: ‘To the philosophies described, there succeeded the work of Plato, which in most respects followed these men, though it had some features of its own apart from the Italian philosophy.’1
To make such a statement, Aristotle had to be fairly confident he knew what the ‘Italian philosophy’ was before it fell into Plato’s hands. His research was extensive and careful, including the work of Philolaus and Archytas and other sources we know little or nothing about, and he recorded the results in several books.[1] Unfortunately, those devoted entirely to the person of Pythagoras and Pythagorean teaching are lost, but because he spent so much time and effort on them, and referred elsewhere to his ‘more exact’ discussions in them, there is no doubt Aristotle knew the subject well.[2] References and quotations from the lost books appear in the writings of authors who lived before the books disappeared, making it possible to peer, indirectly, at a few of the vanished pages.2 The result is a window into what Pythagoreans were thinking and teaching before Plato, helping, at least a little, to circumvent that frustrating impasse, the question of whether what later generations thought they knew about the Pythagoreans and their doctrine was only a Platonic interpretation.
Aristotle was one of the earliest, most dependable sources used by Iamblichus, Porphyry, and Diogenes Laertius. His information went back to shortly after Pythagoras’ death (within about fifty years), but in the books that have survived he never claimed that any specific teaching could or could not be attributed directly to Pythagoras. He also made no distinction between the ideas of Pythagoreans who lived close to the time of Pythagoras and those who lived nearer the time of Plato. He used a Greek form that Burkert says is the equivalent of putting words between quotation marks in modern literature – the ‘Pythagoreans’ – though translating it as ‘the so-called Pythagoreans’ would put too negative a spin on it.
Aristotle wrote that what set both Plato and the Pythagoreans apart from all other thinkers who had lived before Aristotle’s own time was their view of numbers as distinct from the everyday perceivable world. However, the Pythagoreans regarded numbers as far less independent of the everyday, perceivable world than Plato did. At the same time, for the Pythagoreans, numbers were also more ‘fundamental’. If these distinctions seem confusing, they were, even for Aristotle. His difficulty deciding and explaining what the Pythagoreans thought about numbers was not, at heart, a matter of being unable to find out. Rather, he could not think with their minds. The discussion he was insisting on having – about what was more fundamental, more abstract, or more or less distinct from sensible things – would not have taken place at all among the first Pythagoreans. Whether numbers were independent of physical reality, or h
ow independent, were not questions they would have thought to ask.
In his attempt to squeeze the Pythagoreans into Plato’s and his own moulds, Aristotle overinterpreted them and became particularly ill at ease with the idea that all things ‘are numbers’. The Pythagoreans, he reported with chagrin, believed that numbers were not merely the design of the universe. They were the building blocks, both the ‘material and formal causes’ of things. Physical bodies were constructed of numbers. Aristotle threw up his hands: ‘They appear to be talking about some other universe and other bodies, not those that we perceive.’
As Aristotle understood the Pythagorean connection between numbers and creation, for numbers to exist, there first had to be the distinction between even and odd – the ‘elements’ of number. The One had a share in both even and odd and ‘arose’ out of this primal cosmic opposition.[3] The One was not an abstract concept. It was, physically, everything. Aristotle was puzzled by that idea, and unhappy with it.
Odd was ‘limited’; even was ‘unlimited’. As the unlimited ‘penetrated’ the limited, the One became a 2 and then a 3 and then larger numbers.[4] This emergence of numerical organisation resulted in the universe humans know. In Aristotle’s words (he was still rankled by the ‘substance’ of the One):
They say clearly that when the One had been constructed – whether of planes or surface or seed or something they cannot express – then immediately the nearest part of the Unlimited began ‘to be drawn and limited by the Limited’ . . . giving it [the Unlimited] numerical structure.
Aristotle had found that, at least in its broad outlines, the numerical creation of the universe was a pre-Platonic Pythagorean concept. However, he often regarded the Pythagoreans with a frown of frustration, like a professor faced with brilliant students who have disappointed him. Though he was, in fact, not consistent in the way he described Pythagorean ideas about numbers, and was never able to define what he thought ‘speak like a Pythagorean’ and say ‘the One is substance’ meant, it is clear that he feared theirs was a sadly earthbound, material view. ‘The Pythagoreans introduced principles,’ said he, that could have led them beyond the perceptible world to the higher realms of Being, but then they only used them for what is perceptible, and ‘squandered’ their principles on the world itself as though nothing else existed besides ‘what the sky encloses’.3
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